tailieunhanh - Stationary values of the transmission ratio of the planar fourbar

The transmission ratio of the planar fourbar, . the ratio of the angular velocities of input link and output link, is a function of the input angle. Freudenstein [1] showed how to calculate stationary values of the transmission ratio. In the present paper a new method is described. Like Freudenstein’s method it results in a sixth-order polynomial equation. | Vietnam Journal of Mechanics, VAST, Vol. 30, No. 4 (2008), pp. 359 – 365 Special Issue of the 30th Anniversary STATIONARY VALUES OF THE TRANSMISSION RATIO OF THE PLANAR FOURBAR J. Wittenburg Institute of Technical Mechanics, University of Karlsruhe, Germany Abstract. The transmission ratio of the planar fourbar, . the ratio of the angular velocities of input link and output link, is a function of the input angle. Freudenstein [1] showed how to calculate stationary values of the transmission ratio. In the present paper a new method is described. Like Freudenstein’s method it results in a sixth-order polynomial equation. 1. INTRODUCTION The planar fourbar is composed of four links, namely the fixed link of length ` , the input link of length r1 , the output link of length r2 and the coupler of length a (see the fourbar A0 ABB0 in Fig. 1). The angles of rotation of input link and of output link relative to the fixed link, both positive counter-clockwise, are denoted ϕ and ψ , respectively. The transfer function determines ψ as function of ϕ . The time derivative of this function yields an expression for the transmission ratio i = ϕ/ ˙ ψ˙ as function of ϕ . Subject of investigation are stationary values of i(ϕ) . In ref.[1] Freudenstein gave a sixth-order polynomial equation for a certain geometrical variable. The roots of this equation determine the input angles at which i(ϕ) is stationary. In the present paper a new sixth-order polynomial equation with cos ϕ as variable is formulated. It is shown that the coefficients of this polynomial are invariant with respect to an interchange of ` and r1 . From this it follows that two fourbars with interchanged link lengths ` and r1 have stationary transmission ratios imax and imin , respectively, at identical input angles ϕ . Under certain conditions on the link lengths the polynomial is of fifth or of third or of second order. It is not shown in the present paper that also Freudenstein’s equation can be written in a .